graded module
It is a wellknown fact that a polynomial^{} (over, say, $\mathbb{Z}$) can be written as a sum of monomials in a unique way. A monomial is a special kind of a polynomial. Unlike polynomials, the monomials can be partitioned so that the sum of any two monomial within a partition, and the product^{} of any two monomials, are again monomials. As one may have guessed, one would partition the monomials by their degree. The above notion can be generalized, and the general notion is that of a graded ring^{} (and a graded module^{}).
Definition
Let $R={R}_{0}\oplus {R}_{1}\oplus \mathrm{\cdots}$ be a graded ring. A module $M$ over $R$ is said to be a graded module if
$$M={M}_{0}\oplus {M}_{1}\oplus \mathrm{\cdots}$$ 
where ${M}_{i}$ are abelian^{} subgroups^{} of $M$, such that ${R}_{i}{M}_{j}\subseteq {M}_{i+j}$ for all $i,j$. An element of $M$ is said to be homogeneous of degree $i$ if it is in ${M}_{i}$. The set of ${M}_{i}$ is called a grading of $M$.
Whenever we speak of a graded module, the module is always assumed to be over a graded ring. As any ring $R$ is trivially a graded ring (where ${R}_{i}=R$ if $i=0$ and ${R}_{i}=0$ otherwise), every module $M$ is trivially a graded module with ${M}_{i}=M$ if $i=0$ and ${M}_{i}=0$ otherwise. However, it is customary to regard a graded module (or a graded ring) nontrivially.
If $R$ is a graded ring, then clearly it is a graded module over itself, by setting ${M}_{i}={R}_{i}$ ($M=R$ in this case). Furthermore, if $M$ is graded over $R$, then so is $Mz$ for any indeterminate $z$.
Example. To see a concrete example of a graded module, let us first construct a graded ring. For convenience, take any ring $R$, the polynomial ring $S=R[x]$ is a graded ring, as
$$S={S}_{0}\oplus {S}_{1}\oplus {S}_{2}\oplus \mathrm{\cdots}\oplus {S}_{n}\oplus \mathrm{\cdots},$$ 
with ${S}_{i}:=R{x}^{i}$. Then ${S}_{i}{S}_{j}=(R{x}^{m})(R{x}^{n})\subseteq R{x}^{m+n}={S}_{i+j}$.
Therefore, $S$ is a graded module over $S$. Similarly, the submodules^{} $S{x}^{i}$ of $S$ are also graded over $S$.
It is possible for a module over a graded ring to be graded in more than one way. Let $S$ be defined as in the example above. Then $S[y]$ is graded over $S$. One way to grade $S[y]$ is the following:
$$S[y]=\underset{k=0}{\overset{\mathrm{\infty}}{\oplus}}{A}_{k},\text{where}{A}_{k}=R[y]{x}^{k},$$ 
since ${S}_{p}{A}_{q}=(R{x}^{p})(R[y]{x}^{q})\subseteq R[y]{x}^{p+q}={A}_{p+q}$. Another way to grade $S[y]$ is:
$$S[y]=\underset{k=0}{\overset{\mathrm{\infty}}{\oplus}}{B}_{k},\text{where}{B}_{k}=\sum _{i+j=k}R{x}^{i}{y}^{j},$$ 
since
$${S}_{p}{B}_{q}=(R{x}^{p})(\sum _{i+j=q}R{x}^{i}{y}^{j})=\sum _{i+j=q}R{x}^{i+p}{y}^{j}\subseteq \sum _{i+j=p+q}R{x}^{i}{y}^{j}={B}_{p+q}.$$ 
Graded homomorphisms and graded submodules
Let $M,N$ be graded modules over a (graded) ring $R$. A module homomorphism^{} $f:M\to N$ is said to be graded if $f({M}_{i})\subseteq {N}_{i}$. $f$ is a graded isomorphism^{} if it is a graded module homomorphism and an isomorphism. If $f$ is a graded isomorphism $M\to N$, then

1.
$f({M}_{i})={N}_{i}$. Suppose $a\in {N}_{i}$ and $f(b)=a$. Write $b=\sum {b}_{j}$ where ${b}_{j}\in {M}_{i}$, and ${b}_{j}=0$ for all but finitely many $j$. Then each $f({b}_{j})\in {N}_{j}$. Since ${N}_{j}\cap {N}_{i}=0$ if $i\ne j$, $f({b}_{j})=0$ if $j\ne i$. Therefore $b={b}_{i}\in {N}_{i}$.

2.
${f}^{1}$ is graded. If $a\in {f}^{1}({N}_{i})$, then $f(a)\in {N}_{i}=f({M}_{i})$ by the previous fact. Then $f(a)=f(c)$ for some $c\in {M}_{i}$, so $a=c\in {M}_{i}$ since $f$ is onetoone.
Suppose a graded module $M$ has two gradings: $M=\oplus {A}_{i}=\oplus {B}_{i}$. The two gradings on $M$ are said to be isomorphic if there is a graded isomorphism $f$ on $M$ with $f({A}_{j})={B}_{j}$. In the example above, we see that the two gradings of $S[j]$ are nonisomorphic.
Let $N$ be a submodule of a graded module $M$ (over $R$). We can turn $N$ into a graded module by defining ${N}_{i}=N\cap {M}_{i}$. Of course, $N$ may already be a graded module in the first place. But the two gradings on $N$ may not be isomorphic. A submodule $N$ of a graded module $M$ (over $R$) is said to be a graded submodule of $M$ if its grading is defined by ${N}_{i}=N\cap {M}_{i}$. If $N$ is a graded submodule of $M$, then the injection $N\mapsto M$ is a graded homomorphism.
Generalization
The above definition can be generalized, and the generalization^{} comes from the subscripts. The set of subscripts in the definition above is just the set of all nonnegative integers (sometimes denoted $\mathbb{N}$) with a binary operation^{} $+$. It is reasonable to extend the set of subscripts from $\mathbb{N}$ to an arbitrary set $S$ with a binary operation $*$. Normally, we require that $*$ is associative so that $S$ is a semigroup^{}. An $R$module $M$ is said to be $S$graded if
$$M=\underset{s\in S}{\oplus}{M}_{s}\text{such that}{R}_{s}{M}_{t}\subseteq {M}_{s*t}.$$ 
Examples of $S$graded modules are mainly found in modules over a semigroup ring $R[S]$.
Remark. Graded modules, and in general the concept of grading in algebra^{}, are an essential tool in the study of homological algebraic^{} aspect of rings.
Title  graded module 
Canonical name  GradedModule 
Date of creation  20130322 11:45:05 
Last modified on  20130322 11:45:05 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  18 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16W50 
Classification  msc 74R99 
Synonym  graded homomorphism 
Synonym  homogeneous submodule 
Related topic  GradedAlgebra 
Defines  graded module homomorphism 
Defines  homogeneous of degree 
Defines  graded submodule 
Defines  grading 